[130] PROFESSOR DE MORGAN, ON SOME 



f(V w - V p ' + ...) Qdx. This, it is assumed, must vanish; which, though clear enough in the 

 common case in which Q = y, and V y — ... is a function of a? only, is not sufficiently supported 

 in any other. Why may not (V y — ...) Q be itself a new integrable function? 



Let V be an integrable function, say of x, y, p, q, r, a, t, which therefore can be thrown 

 into the form 



V = X + Yp + Pq + Qr + Rs + St, 



where X, Y,...S are functions of w, y, p, q, r, s, and not of t, and are themselves differential 

 coefficients of fVdx. Consequently we must have all such relations as P r = R , X q = Q x , &c. ; 

 or, M and N being any two of the capitals, and m and n their corresponding small letters, we 

 must have M n =N m . If then we differentiate with respect to any one letter, say r, we have 



V r = (X T + Y r p + P r q + Q r r + R r s + S r t) + Q, 

 = (72, + R v p + R p q + R q r + R r s + R,t) + Q, 

 = R'+Q. 



Proceeding thus, we get* 



V X = X', s=v t , 



V^Y' + o, R=V S - v;, 



v p = p j +y, Q =Vr -r;+v;\ 



v 9 =q' + p, p= r, - v; + vr - v;\ 



r r -R' + Q, y=v p - v; + v r " - vr + v; v , 



v, = s' + r, o - v y - r/ + v q " - vr + v; y - v t v , 

 V t = o + s. 



whence the necessity of the criterion is established. (The general theorem on which it depends 

 is 



{W)^= (WW+ W^. n (W), 



which, we may notice in passing, gives 



iW»)^ = (w yW ) w + k (fv/- 11 + kk -~ (WW" 2 ' + - 



the last term containing (W^"~ n) , whenever k is greater than n. It follows that (W m ) w is 

 always integrable per se k —n times, whenever k is greater than n. 



The last six equations give the first: for if we differentiate V= X+ Yp + Pq + ... 

 we have 



V, + V y p + V p q + ... = X'+Y'p + (P'+Y)q+ 



which, by the last six equations, is V x = X'. And thus it appears that the partial differential 

 coefficient of X + Yp + ... with respect to any letter (except only w) is so much of the total 

 differential coefficient as is seen in the explicit coefficient of the next letter. Again, we must 

 observe the following theorems : 



» I cannot anywhere find it noticed that the well known I coefficients of the integral required. [Except by M.Sarrus.J 

 forms V,, V, - V,', &c. are nothing but the partial differential ' 



