546 Mb DE MORGAN, ON SOME POINTS 



Ampere has noted that S 2 = 4,(RT + QU) is a condition necessary in order that 



Q + Rr + &c. m 



may have a primitive containing two arbitrary functions of one variable, such as we obtain 

 from the polar form (p (x, y, », a, b, v) = 0, by assuming b = Xa, c a fxa, and obtaining a from 



The preceding investigation will sufficiently illustrate the restrictions under which equa- 

 tions of three variables are analogous to those of two. In the latter case, there is always 

 coexistence of intermediate primitives, but not always so in the former : and the former can- 

 not be treated co-ordinately with the latter except in the cases in which intermediate primi- 

 tives coexist. 



24. The notation for differential coefficients which I have now used for more than 

 twelve years, with the extensions here proposed, will bring this paper into half the space 

 which it would have filled, if all the formulae had been written in the usual way. But this is 

 not the only advantage. The equation which I have written as 



(JB m + (AB m r + \{AB W) + (ABj s + (AB xq) t + (AB„ (#»- rt) = 0, 



would, written in the ordinary way, have been 



IdA dA \ IdB dB \ idA dA \ IdB dB \ 

 \dx dm ) \dy dx J \dy dsx J \dx dss J 



(dA IdB dB \ _dB^ldA dA \ 1 

 [dp \dy dss I dp \dy dz Jj 



dB \ dB (dA dA \ dB (dA dA \ dA (dB dB \] 

 - + ^ q )-lla-{Ty + ^ q ) + lp-{lix + di; P )--dp-{-d*- + d*- p )\ S 



(dA dB dA dB 

 ^dq dp dp dq 



dA idB 

 dq \ dy 



[dB (dA dA \ dA (dB dB \] (dA „ 



the first containing 55 types, without the parentheses 43, the second 258. And it would have 

 been exceedingly difficult to have made the common form of the coefficients an object of 

 thought, if the coefficients had been either detailed or represented by single letters. 



The adoption of occasional- symbols in abbreviation of formulae, though often necessary, 

 naturally belongs to cases in which the forms are not of frequent occurrence. In some few 

 cases, as in the well known use of p, q, r, s, t, an occasional notation has become permanent : 

 but this cannot often happen. The want of some system of abbreviation drives many writers 

 to a frequent adoption of occasional symbols. For example, the following passage occurs in 

 the proof which M. Cauchy (Moigno, Vol. u. p. 447) gives of his remarkable theorem already 

 mentioned. " Representons toujours par F(x, y) = C Fintegrale generale de l'equation 



dy=f(x, y)dx, 



