JIr. ELLIS, ON THE METHOD OF LEAST SQUARES. 213 



I proceed to show that in a particular case in which the value of P can be accurately 

 determined, Laplace's approximation is correct. It has sometimes been thought that the intro- 

 duction of the negative exponential involves a petitio principii, and is equivalent to assuming a 

 particular law of error. It is therefore desirable, and I am not aware that it has hitherto been 

 done, to verify his result in an individual case. 



Let the law of error be the same in all the observation.s, and such that d)e = ie'", the upper 

 sign to be taken when e is positive. 



Let /x, = /u, = &c. = 1, then 



P = -f''(laj'"'(l€^'<)d6, y ''"(ic"")df„cosa(?^ - le). 



1 r" cos Ma , ■ r" 



/J = — / 7:r "Of since / e' cos a€de = 



TT -/„ (1 + aO° -^0 



1 +a^ 



The value of p is thus given by a known definite integral, which has been discussed by M. Catalan 



in the fifth volume of Linumlle's Journal. 



It may be developed in a series of powers of «/. Up to u"-^" - '* no odd power of u can appear 



r" a}' 

 in this development, for / „- da is finite while p is less than n, and therefore the integral 



may be developed by Maclaurin's theorem. For higher powers the method ceases to be applicable, 

 and we must complete the development by other means. But as we suppose n to increase s. 1. the 

 integral tends to become developable in a series of even powers only of?/. Thus 



f" cosua , /•» da ^ ■> r" «' 



/ , o,- da = / , 5- - kn^ / ^ da + &c. 



( 

 , /• » da 



Jr'^ a' da 



Then 

 and generally 



Now 



-x^U 





... IT 1 . 3...2TO - 3 



f(n) = -. ; 



2 2.4...2W - 2 



. „, , TT 1 . 3...2n - 5 



- ^f{n - I) =-. . 1, 



2 2. 4.. .278 - 2 



AS/-/ ^N ■^ ' .3...2W-7 



AY (« - 2) = - . .1.3; 



•^ 2 2.4....2W -2 



!ind generally. 



■ A„,/ ^ "^ 1 .3...2W -3 -2p 



■'^ '^' 2 2.4...2«-2 '^ 



1 .3...2p - 1 



= /(«) ■ — ^ - • 



%n- \ -2p,.,2n - S 



EB 2 



