396 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS, &c. 



Also, cos /.pPs = a == , and 8X = — s/\ + a* + b\ Hence 



Vl + dr + o 2 a 



V VbV ad IV 



Iw r l + r <>r 



%X v/(l + d + b") (1 + d~ + V*) ' 



If therefore the variation of velocity bV from P to p be the 

 same as from P to s, that is, if the variation be nothing from p 



to s, the limit of the ratio -5— , is the differential coefficient -y- , 



dr dr 



taken as if the variation were from one point to another of the line 



OWP. Hence, 



V Vbb' , dV 



t , — + -j +aa .— j- 



s dw r I + r dr 



dX v/(i + a * + b*) (1 + a' 2 + b") ' 



The value of -r~ is evidently derived from that of -pp. by in- 

 terchanging a, a, and b, V ; and since, when this is done, the right- 

 hand side of the equation is unaltered, we conclude that -r^v = -r>s- 



d X dZ 



~ dw dv , du dv 



So -dT = dz' and dY=dX- 



Consequently, udX + vdY + wdZ is an exact differential. 



Cambridge Observatory, 

 May 27, 1842. 



