dy dt F(1) = F()=-£fp), F'(d) =ƒ'(v), F'(x) =ƒ′(x), F'(y)=f'(y), F'(t)=f'(t); therefore (6) gives us 0=ƒ'(p) + (6) V±√V2+4Rp2 2p2 ·ƒ'(v); · (7) If w1=C, and w2 = C2 be the respective integrals of the equa tions 2 we shall have in w1, w, two values for f, since they obviously satisfy (7) and also, being free from x, y, and t, satisfy (8). Again, we have for the integration of (8) the following auxiliary equations, viz. Hence we shall have the two following additional values for f, viz. where 2p (9) is arbitrary, provided these values satisfy also equation (7). But if (9) satisfies (7), the latter will be satisfied by X f=$ { x_ V = √ V® +4Rp2 . 1 }, in order to which we must have d 2D 0 = dp (V = √ V2+4Rp3) d 0= (V + √ V2+4Rp2) dp (V ‡ ✓ V2+4Rp2) (10 a) which, taking alternately the upper and lower signs, constitutes a pair of equations of condition which must be satisfied when f is susceptible of both the values (9). Moreover, since if (9) satisfies (7) the latter will be satisfied by f=&{y. (V + √ √2 + 4 Rp2 + v) t }, 2p a conclusion which may be rejected on account of defect in generality*. Therefore when equations (10) hold, (9) must reduce itself to f=¢{ (x_V+ √ √2+4Rp2. t), p, v},.. (11) 2D and we shall thus have for f four values in all, viz. those given by the last equation together with those previously found, viz. W1, W2. Now it will be remembered that the original integral which we assumed for (2) was (2 a), or, which is the same thing, It is clear, however, that this last equation may be put under the form fa(xytpv)=x_{f(xytpv)}, where X-1 denotes an arbitrary function; and if we treat this equation in the same way in which we have treated (12), we shall arrive at precisely the same formulæ for the determination * The grounds upon which I rest this conclusion will hereafter be more distinctly pointed out. off as those at which we have already arrived for the determination of f. Thus the occurrence of two of the four values of which ƒ has been shown to be susceptible is at once accounted for; and if we reflect that almost universally when an equation of the second order is integrable by Monge's method, there are two equations of the first order of the form (2 a) from each of which it is separately derivable, there cannot, I think, be a doubt that the four values above derived for f must be attributable to the values of f and f in equation (12), and to the corresponding functions in the other first integral of (2). Acting from this clue, we shall be justified in assuming, and we shall find it to be the fact, that when the equations of condition (10) are satisfied, (2) will be susceptible of the two following integrals, from each of which separately it is capable of being derived, viz. If we now recur to the equations of condition (10) or (10a), or, as they may be written, P1 =V+ √V2+4Rp2, P2=V− ✔ V2+4Rp2, (14) the form of the equations leads us to conclude that, if neither of the quantities P, P, vanishes (as for instance P1) so as to render dP, dᏢ = =0, we shall have dp dv 2 which implies that we have V2+4Rp2=0,-a conclusion which I have aleady pointed out as one to be rejected as deficient in point of generality*. This may be seen more distinctly as follows. (13) become In the case supposed, whence we get = funct. w; and therefore we must have p = funct. of v, since we involve p and v only. Now not only is this conclusion, viz. that p = funct. v, clearly defective in point of generality, but, if it were true, we But if one of the quantities P1, P, is constant-for instance, if we have 2 V+ √V2+4Rp2=2a, where a is constant, we shall have a value which, it will be found, satisfies both the equations of condition (14). In this case equations (8 a), from which w1, we are to be derived, become 0=dv a {x (v + 2) - a} do ; .. w1 =v+, and equations (13) become a ρ (15) If we put u v +, the three equations which determine the circumstances of the motion, i. e. the pressure, density, and velocity, become might assume p= a function of p only--an assumption which is open to all the objections which in my former paper have been shown to apply to the law of the received theory, viz. p=a2p. |