A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter.

  • Isaac Todhunter
Date:
1865
Catalogue details

Licence: Public Domain Mark

Credit: A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter. Source: Wellcome Collection.

    542/648 (page 522)
    and between each of these values it will be found that the ex- pression is numerically a maximum, and it is also a maximum when cp = 0. Thus we may calculate by Art. 957 the value of the integral [(snmmM when the limits are consecutive multiples of —. J\suxcpJr m The equation which determines the maxima values of -1? A sin cp m cos mcp sin cp — cos cp sin mcp _ A ; 2 i sm cp is It will be found that this is satisfied when (p = 0 ; the situation of the other values of cp will be more easily discovered by putting the equation in the form tan mcp — m tan cp = 0 : 5tt now we see that the next solution will lie between vup = — and (£ = ^, and then the next between mcp - ^ and mcp = ~, m and so on. We proceed then to find l sin mcp sin cp dcp. The maximum value of the function which is to be integrated occurs when cp = 0, and is therefore m*; assume sin mcpy —:—A ] = ms e 1 sm cp therefore mcp — ^ mscpa + ... <f>-qP + ••• = mae 0 take logarithms, thus we obtain — 1) </>’ + ...