METHOD OF TRANSFORMING AND RESOLVING EQUATIONS, 335 



Repeating this process, we find, next, the condition, 



m-3> h'\ + h'\ + A"'3 + . . . + h'\, . . . (239.) 

 and generally 



m-e>M') + aW + A(») + ... + A^'); • . . (240.) 



each new condition of this series including all that go before it, 

 and the symbol A^*) being such that 



h(^ = h, 



and 



A('+i)-A»=-l, 



(241.) 

 (242.) 



(243.) 



t — n t — n t — n -\- \ 



Integrating these last equations as equations in finite differences, 

 we find 





= A,_2 + i A,-i + i . ^i . {ht - '-^) 



^t-2 - "t-2 



f'f_. = K-3, + i ht_2 + i ^- A^_i 



+ 1 



1±J:L±J (} 

 2 3 V 



i + 3'' 



)■' 



> (244.) 



hf = ^ + ^ A, + i L±i A3 + e ^-1 ^l±-2 A, + . . , 



■ ^•+l^^ + 2 ^• + ^-2/ z + Z-i y 



2 3 ^— 1 V ^ t J _ 



And making, in these expressions, 



i-hf, 



so as to have 



A«=o, 



and putting, for abridgement, 



h^\) = ^Ai, Ip = ^Ag, . . . A;^)^= ^A,_i, 



(245.) 

 (246.) 



(247.) 



