A new contribution on the problem of plot size in experiments with trees

Em trabalho anterior, introduziu-se novo método de estimação do tamanho ótimo de parcelas experimentais para plantas arbóreas. Esse método, que leva em conta as bordaduras e utiliza o coeficiente de correlação intraclasse (p) entre árvores úteis dentro das parcelas, define como tamanho ótimo o número k de árvores úteis que minimize a variância da média de cada tratamento, para um número total de árvores (N), considerado fixo. No artigo referido, esse tamanho ótimo foi determinado, para parcelas com meia-bordadura ou com bordadura completa, com uma ou com duas linhas de árvores úteis. No presente artigo, o problema é estudado de modo mais geral, buscando o tamanho ótimo quando varia tanto o número de linhas úteis (n) como o de árvores úteis (k) por parcela. No caso de meia-bordadura, o número ótimo de linhas úteis é n =  [(1 - p)/p]0,333, e no caso de bordadura completa, n = [2(1 - p)/p]0,333, sendo sempre k = n2, p>0. Por outro lado, demonstra-se que, no caso de bordadura completa quando se passa de um ensaio com k árvores úteis em n linhas, com N árvores totais por tratamento numa área A, para outro semelhante com essas características indicadas por k', n', N' e A', sem mudar a variância da média de cada tratamento, temos: A'/A = N'/N = {(1 + 2/n') (1 + 2n'/k') [1 + (k' - 1)p]}/ {(1 + 2/n) (1 + 2n/k)[1 + (k - 1)p]}.In a previous paper, a new method was proposed for the estimation of the optimum plot size of experimental plots for trees. The method, which takes in consideration the guard rows and uses the intraclass coefficient of correlation (p) among test trees within plots, defines as optimum size the number k of test trees which minimizes the variance of the estimate of a treatment mean for a fixed total number (N) of trees per treatment. In that previous paper the optimurn size was determined, for plots with one half or complete guard rows, with one or two test rows. In the present paper, the problem is generalized, the optimum size being searched when both the number of test trees (k) and the number of test rows (n) are variable. For the case of one half guard row the optimum number of test rows in n = [(1 - p)/p]0.333, and for the case of a complete guard row is n = [2(1 - p)/p]0.333, always with k = n2, p>0. On the other hand, it is shown that, with complete guard rows, when changing a trial with k test trees per plot in n rows, with total number of trees per treatment N and area A, into another one with these parameters indicated by k', n', N' and A', keeping constant the variance of the estimate of a treatment mean, we have: A'/A = N'/N = {(1 + 2/n') (1 + 2n'/k') [1 + (k' - 1)p]}/ {(1 + 2/n) (1 + 2n/k)[1 + (k - 1)p]}

A lei de Mitscherlich aplicada a experimentos de adubação com vinhaça

The author studies, with the aid of Mitscherlich's law, two experiments of sugar cane fertilization with vinasse. The first one, carried out in Piracicaba, State of S. Paulo, by ARRUDA, gave the following yields. No vinasse 47.0 tons/ha. 76.0 tons/ha. 250 c.m./ha. of vinasse 75.0 do. 112.0 do. 500 do. 90.0 do. 112.0 do. 1000 do. 98.0 do. 107.0 do. Data without NPK were appropriate for the fitting of the law, the equation of which was found to be: y = 100.8 [1 - 10 -0.00132 (x + 206) ], where y is measured in metric tons/hectare, and x in cubic meters/hectare. The optimum amount of vinasse to be used is given by the formula x* = 117.2 + 1 log w u , ______ ____ 0.00132 250 t being u the response to the standard dressing of 250 cubic meters/hectare of vinasse, w the price per ton of sugar cane, and t the price per cubic meter for the transportation of vinasse. In Pernambuco, a 3(4) NPK vinasse experiment gave the following mean yields: No vinasse 41.0 tons/hectare 250 cm./ha. of vinasse 108.3 do. 500 do. 134.3 do. The equation obtained was now y = 150.7 [1 - 10 -000165 (x + 84)], being the most profitable level of vinasse x* = 115.2 + 1 log w u , _______ ____ 0.00165 250 t One should notice the close agreement of the coefficients c (0.00132 in S. Paulo and 0.00165 in Pernambuco). Given the prices of Cr$20.00 per cubic meter for the transportation of vinasse (in trucks) and Cr$ 250.00 per ton of sugar cane (uncut, in the fields) the most profitable dressings are: 236 c.m./ha. of vinasse in S. Paulo, and 434 c.m./ha. in Pernambuco

Dois teoremas sôbre a função gama

This paper proves the following theorems on the gamma function: Theorem I The integral &#8747;O&#8734; t u e-t dt = &#915; ( u + 1 ) , where u, real or complex, is such that R (u) &gt; -1, will not change its value if we substitute z = Q (cos &#966; + i sen &#966;) for the real variable t, being jconstant and such that - &#928;/2 < &#966; < &#928;/2 , Theorem II The integral &#8747;-&#8734;&#8734; w2u + 1 e -w² dw = &#915; ( u + 1 ) , where 2u + 1 is supposed to be a non negative even integer, will not change its value if we substitute z = w + fi, f being a real constant, for the real variable w. The proof of both theorems is obtained by means of the well known Cauchy theorem on contour integrals on the complex plane, as suggested by CRAMÉR (1, p. 126) and LEVY (3, p. 178)

A estimação do efeito residual de fertilizantes por meio da Lei de Mitscherlich

This paper deals with the estimation of the residual effect of fertilizers through the use of Mitscherlich's law. The formulas and reasonings now presented are a further development of those introduced previously by PIMENTEL GOMES (2). The new formulas allow the estimation of the residual effect h in cases where the experiments are carried out in the same plots for two or three subsequent years (or crops). In an experiment analysed as an example, the residual effect of calcium hydroxide was estimated to be h = 0.423, that is, about 42%, so that one should advise the use of frequent application of small amounts of lime instead of heavy quantities used at long intervals