518 Mr DE MORGAN, ON SOME POINTS 



A relation such as u (x, y) = co , gives the same form for y as u = const, generally ; 

 namely y = — u x :u y . If any doubt should arise as to our right to make c = oo in u = c, 

 or any question as to distinction between infinite constants and infinite variables, we may have 

 recourse to «"' = 0, which gives y = - u x :Uy. 



4. If M = 0, or M = oo , preserve P finite, then in (MP) X or ifefP, + PM„ MP X is 

 infinitely small compared with PM X : the ratio is that of (log P) x to (log M)„ log P being 

 zero or finite, while log M is infinite ; so that the theorem in ^ 3 applies, and proves the 

 assertion. Generally, M = and M = oo give to MP, and PM X the ratio of log P 

 to log ili". 



Let Pdx + Qdy be made integrable by the factor M, so that (MP) y = (MQ) X . If M a 

 or co make P and Q both finite, the last gives PM y :QM x =1 or P:Q::M x :M y . But 

 J/ = and if = co , being cases of JW = const., both give y = - M x :M y ; whence both are 

 solutions of P + Qy' ■ 0. Hence the well known properties of the integrating factor ; but the 

 extreme cases offer some difficulty, which may be avoided as follows. Let U be a function of 

 x and y, then U = const, (co inclusive) gives y = - U x :U y , and £7, = const, and U y = const, 

 (co included) give y = - Uj.U^ and y = - U^.U W . Now if a relation between y and 

 a? make U x and t/ y feo^A vanish, or make either or both infinite, the three forms of y' are 

 either zeros, finite and equal, or infinites : so that the selected relation satisfies U z + U y y = 0, 

 the differential equation of U = const. 



This reasoning might easily be extended to equations of n variables ; and most easily by 

 making n - 2 of them indeterminate functions of the (n - l) th . The want of such consider- 

 ations is rather a defect in the elementary treatment of differential equations. For example, 

 let U= U{x,y) give dU m U y (Pdx + dy). The differential equation of U= const, is at 

 once taken to be Pdx + dy = 0, be the constant what it may. But may there not be singular 

 exceptions, in which U = a specific const, produces dU = 0, not through Pdx + dy = 0, but 

 through U m ? The answer is, that U = 0, P not being infinite, is accompanied by 

 Pdx + dy = 0, as shown. 



5. In my last paper I pointed out that the differential equations intermediate between 

 the original primitive and the one which has lost all the constants cannot be represented with- 

 out the aid of arbitrary functions. Lagrange, I have since observed, has noticed this exten- 

 sion, and has rejected it on grounds which I shall presently discuss. The ordinary theory is 

 complete as to the passage from the original primitive to the final equation, upon the suppo- 

 sition that all relation between the constants, and between those constants and others, is 

 denied or forbidden. But it is neither true nor complete as to the passage from the final 

 equation to the original primitive. 



Let xy = ax + by + c. So long as a, b, c, are wholly unconnected, no primordinal equa- 

 tions can be deduced except 



y +xy' = a + by', y* = a(y - xy') - cy', w*y = b (xy- y)-c; 



