344 REPORT— 1869. 



It is easily seen that if 



- "^K 



/2K 



fP"%(.v) _(_y.m r7»-\/V ^ ( _ -1 )»■ ^^^ ^ A . 



d.^'' 



(//i" 



(-1)" 



f/A" 



0,(.r)=VA-^^^^^ 



(/A 



.r= r?VA 



1.2 dh- "1.2.3.4 



+ ; 



(1) 



and therefore, putting B= ^lL i ' — ^ — , 



1" Jo Vl— ^^sin^^ 



a=4(l-2P), h=2l"k'\ 



we obtain, by a process similar to that used in obtaining equations (1) and (2) 

 of last article, 



^=2A% '^=hA.\ ^=-166A=, '^=ahA.' . . (2) 

 ah dh dh dh 



If H is a function of A, B, a, h, M'e obtain from these equations the fol- 

 lowing : — 



dh 



.cm 



dE 



=2A-B'-V:jt + bA.' ^I^ +«6A' 



c^A 



f?B 



.(IK 



dh ' 



■imd 



dK 



da 



(3) 



Differentiating Va a certain number of times in accordance with (3), we 

 are led to suspect that 



2m + I 



2m+l 



^^!^=n/'»)A"T" B'" + nJ'«',.^A"^' +=^B 

 dh'» ' ' - i 



2m+l , „ i„.T-. , . 



1 v'»») i ^i; 1-3 -pm — 3 . (m) . — r — + >-n»J-l , 



+ n3 VA 2 ±i ^ . . -|_H> ^r,A ^ B +. . . ^ 



jm-2 

 2m+l 



(4) 



2JH + 1 



/ „, \ : + in 



where n/'"^ '"./'"^ &c. are certain integers, i\, r^, &c. certain functions of a 



2>n + l 



+ i+p m+p—i 



and h independent of (»;) ; so that the coefficient of A ^ B in the 



expansion of ^ is vr"' "'"-'' ''?•,•. It will be observed that ri = 0. 



Furthermore, we are led by induction to suspect that the numbers n ' are 

 6uch that 



-UB*- 



A^r' 



e^O^O^VAe- ■|l + '-'^r:2T3T4''''u.2.3.4.5.6 

 These assumptions may be tlius verified ;— 



+ ... . . (5) 



