LAGEAXGE. 303 



This may be established by differentiation. For thus 



that is 



(a+cz) {l-{-A^z + A,/ + .., +Ay+ ...} 



= (l-2a2-cs') [Aj^ + 2A^z + .,, +nA^z''~'-^ ...}; 

 then by equating coefficients the result follows. 



559. In the second problem the same suppositions are made 

 as in the first, and it is required to find the probability that the 



error of the mean of n observations shall not surpass + — . 



Like the first problem this leads to interesting algebraical ex- 

 pansions. 



We may notice here a result which is obtained. Suppose we 

 expand {a + 5 (ic + £c"^)}" in powers of x\ let the result be de- 

 noted by 



A^ + A^ (x-^x-') +A^ {x'^x-') -{-A, {x'+x-^ + ... ; 



Lagrange wishes to shew the law of connexion between the co- 

 efficients Aq, A^, A^, ... This he effects by taking the logarithms 

 of both sides of the identity and differentiating with respect to x. 

 It may be found more easily b}" putting 2 cos ^ for a? + x~^, and 

 therefore 2 cos rO for a?*" + x~'\ Thus we have 



(a + 25 cos ey = ^0 + ^A cos 6 + 2A^ cos 26 + 2A^ cos 8^ + . . . 



Hence, by taking logarithms and differentiating, 



Tib sin 6 _ A^ sin 6 + 2A^ sin 26 + ^A^ sin 3^ +. . . 

 a-^2bco^6~ A^ + 2A^ cos 6 + 2A^ cos 2^ + . . . 



Multiply up, and arrange each side according to sines of mul- 

 tiples of 6 \ then equate the coefficients of sin r6 : thus 



rib [A,_^ - A,^,] = raA, + b [{r - 1) A,_^ -f (r + 1) A,^,] ; 

 therefore A^^, = b {n-r^rl) A -raA^ 



