14^2 CELESTIAL MECHANICS. 1. ii. 12. 



tions for finding the value of -j- : but it will be sufficient 

 ^ d^ 



at present to consider that part which is multiplied by v —1, 



, , . , ^ , , . mt . du ^ /m . 



and which affords us the equation e^ sin y if— + e 2 ^— sm 



mt 



yt—y cos yt)ur:z — ife T sin yt u'dit — . . . ; the flowing 

 quantities in the second member being supposed to begin 

 with t. Now, at the end of the ascent, putting the time 

 T, the fluxion d* vanishes, and with it dw, which is —{nu 



-\-l)ds; at this moment, then, we have e ~T u (—sin 7 T — 



mt 



y cos yT)-=. — Ife ~2 sin yt u^dt — . . . ; which being uni- 

 versally true, it must be true also when the whole value of 

 u is evanescent, and since in this case u^ is infinitely small 

 in comparison with u, the whole of the fluents in the second 

 member of the equation, which depend on the powers of w, 

 must vanish in comparison with the first member, and we 



shall have Ozz— -sin yT—y cos T, and-r-. — yl'—y.oY tanar 

 2 2 cos ° 



, 2y . 



yTzz — , T being the whole time of describing the arc 5, 



whatever its length may be, since, by the conditions of the 

 problem, this time must always be the same, so that the 



equation 0=: "77 ^in yT — y cos yT will be true in all cases, 



mt 



whence in general — Z.^^ e ^ gin yt u'At — ... = ; but 

 when s and u are small, the first term is the only one that 

 remains considerable, the others vanishing in comparison 

 with it, consequently this term must also vanish, which can 

 only happen if /=:0, since none of the quantities concerned 

 change their values from positive to negative within the 



