STORY. — A NEW GENERAL THEORY OF ERRORS. 189 



that is, the ^*s wiih even suffixes are all derived from A^ and all the A's 

 witli odd suffixes from Ai by the application of definite polynomials in T. 

 "We may, then, determine the forms of the ^'s with even suffixes and of 

 those with odd suffixes separately ; that is, we may suppose our calcula- 

 tions restricted to a degree of approximation that corresponds to a devel- 

 opment of 6(s) in which occur even powers of s up to s-" and odd powers 

 up to s-^+i, without assuming any particular relation between a and (3. 

 Then (72) holds for 1 ^ i ^ a, (73) holds for 1 ^ i ^ fS, while (71) is 

 expressed by 



(T+a)' A>a = 0, (T+ /? + ^) • Ao^+i = 0, 



that is, when A-2a and -^2^+1 have been determined by (72) and (73), 



respectively, 



(74) (T+ a)(-+i) ■ A, = 0,(T+ fS+ i)('3+') • Ji = 0. 



The only conditions imposed upon ^„ and A^ so far are, then, that they 

 shall be linear functions of the P's, with constant coefficients that satisfy 

 the equations (74). The oidy linear functions of the P's that are of 

 a given weight iv are constant multiples of /*„„ while, by (54), 



Ml _ 1 \ 0+1) 

 (T+ /?_+ i)0+l) ■P,,,= [fS- "^ ] P^; 



therefore, in order that A^ (as a linear function of the P's) shall satisfy 

 the first equation (74) it is necessary and sufficient that the suffix w of 

 any P involved in it shall satisfy the equation 



2/ =»• 



that is that w shall be an even integer not greater than 2 a ; and, in order 

 that Ai (as a linear function of the P's) shall satisfy the second equation 

 (74) it is necessary and sufficient that the suffix w of any P involved in 

 it shall satisfy the equation 



that is that w shall be an odd integer not greater than 2/3+1. We 

 may, then, write 



(75) 





 where the C's are constants. 



Ao — ,,^^,.2* P2*> Ai ^^,. 2i+l Pai + i! 



