POINTS OF THE INTEGRAL CALCULUS. 



[133] 



y(v P - r q '+ C- r,'"+rr) + P (r q - v; + v,"- r?') 



+ HK- v:+ V t ") + r (V a - V t ') + sV t 



+ MV r;+ r t "- r;"+ r; - v t ")dx. 



The sufficiency of the criterion now appears again ; and when it is satisfied, we see that we 

 have only a determinate function of % to integrate, and after integration, and addition of V g , 

 to write y — u for y, p — u for p, &c. 



In Poisson's process, the result is that of the preceding with u = : but, as M. Moigno 

 remarks, this supposition may make F infinite or incalculable. 



But the easiest way of arriving at fVdx, in any case that can occur, is by observing that, 

 when the criterion is satisfied, V, and V y are both integrable per se, and give (fVdx) x and 

 (fVdy) y by the process. Again, if t be the last letter that enters, we know that fVdx = fV t ds 

 (s only varying) + (a function of precedents of s) : and if observation be not now sufficient to 

 detect that function, we have* other partial differential coefficients of the integral sought in 

 V — V t ', &c. M. M. Bertrand and Moigno have both preferred an indirect mode, (which in- 

 troduces arbitrary constants to be determined during the process) in preference to the operation 

 1 

 f dx : and the example they have both chosen is 



V = 2% + y a + 2asyp + xq + a?r —p. 

 With this I should proceed as follows: — If jVdoo m U, we have 

 V, = 2 + 2yp + q + Zwr, U x = 2a? + y 2 +p + 2<vq - 2p = 2x + y 2 + 2xq — p, 



V„ = 2y + 2xp, U v = Zxy. 



and a?q must be the only term involving q ; whence the integral must be immediately seen 

 to be 



U = x 2 + xy* + x*q — px. 

 It is sufficiently well known that, in mathematics, undefined dependence is independence, 

 though ' perhaps the maxim never finds words. But it appears in the preceding that this 

 remains true to the following extent. If 6 be any function of a, and if c, e, &c. be the succes- 

 sive differential coefficients of b with respect to a, then, till the connexion of b and a is defined, 

 a, b, c, e, &c, are perfectly independent of each other. This might easily be established 

 otherwise. 



Section V. 



On the demonstration^ of the mode of transforming multiple integrals. 



I have always been dissatisfied with the manner in which Lagrange, Legendre, and all 

 other writers who have treated the subject, establish the mode of changing the variables in a 



* We have them, because the easiest way of establishing 

 the criterion (which must be done first) is by going through the 

 successive formation of 5= V n R= V,— S', Q= V,— R , &c. 



t This section was a subsequent communication, dated 

 February 22, 1851. 



