212 THIRD REPORT— 1833. 



also from this equation that if the values of the transcendent 

 r (;/•) can be determined for all values of r which are included 



(Fonctions Elliptiques, torn. ii. p. 415,) a restriction which is obviously unne- 

 cessary. 



There are two cases in which the coefficient of a;"-'' in the equation 



d'- a" r (1 + M) 



dxr — r(l + n — r) 

 requires to be particularly considered : the first is that in which this coeffi- 

 cient becomes infinite ; the second, that in which it becomes equal to zero. 



The numerator F (1 + n) will be infinite when n is any negative whole 

 number ; the denominator F (l + w — r) will become infinite when n — r is 

 any negative whole number, and in no other case: if w be a negative whole 

 number, and if r be a whole number, either positive or negative, such that n — r 



is negative, then the coefficient fr/^ — ; ^ becomes finite, in as much as 



° 1 (.1 + M — r) 



r (— <) (if ^ be a whole number) = ^— ^ 1( — iV ' ^"^^ ^ ^°) disap- 



d-^ 1 ... 

 pears, therefore, by division : thus all the coefficients of s _y • — are mfinite, 



unless r be a negative whole number, such as — m, in which case it becomes 

 1 . 2 . . m . ( — 1)"', a result which is easily verified. In a similar manner it 



would appear that the coefficients of- , _^ • — are infinite, when m is a posi- 

 tive number, unless r be a negative whole number equal to, or greater than, n. 

 The coefficient jr->Yq; — ^^—\ will become equal to zero, when 1 + w — r 



is, and when 1 -|- w is not, equal to zero or to any negative whole number ; for, 

 under such circumstances, the denominator is infinite and the numerator is 

 finite. 



As the most important consequences will be found to result from these 

 critical values of the coefficient of differentiation, we shall proceed to examine 

 them somewhat in detail. 



(1.) The simple differentials or differential coefficients o{ constant quantities 

 are equal to zero, whilst the difiFerentials or differential coefficients to general 

 indices (positive whole numbers being excepted,) are variable. 



Thus 



,4 J J F (h ' Jit x' A ^-* 



«(ri) 



d^ dx^ ^ (.5J V^TA- <^a; 



^fl+i _ _ . ^_ - ,^_^ , a ^0+1 - ax; 



— F(^) • " - Vt • dx-^ - T (2) 

 and similarly in other cases. 



(2.) The differentials of zero to general indices (positive whole numbers 

 being excepted) are not necessarily equal to zero. 



Thus, if we suppose 



C_ _^ £_0 C_ F (1 — w) ^-„-r. 



a _ _ J, ^Q^ a; », we get ^ ^,. — j, ^^^ " r (l — « — r) ' 



if M be a positive whole number, F (1 — m) := oo , and this expression is finite un- 

 less F (1 — n — r) = 00, in which case it is zero : if r be also a positive whole 



