366 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS 



a' dp a'(f> df ldti\ 



du St d f 



and to the first approximation, — = - a <h -^- . Hence 

 ^^ dt ^'dx 



u = - a' — j(b dt + c, 

 dai 



the arbitrary quantity c being in'general a function of x, y, and z. In the same manner, 



df 

 V = - a' -— f(h dt + c . 

 dy ^' 



., . d'dp .j.d<b ldw\ , . . . dw 



Also since — -' = - a f — ~ = ( — we have to the same degree of approximation, — 



pdn •' pdz \dt I ' ^ *^*^ dt 



= - a'f ■ -~ , and w = - a f I —^ dt + c" = - d-f — '-^^ — + c". But it is evident from the 

 dz ' J dz dz 



assumed law of the condensation in any plane perpendicular to the axis of z, that the accelerative 



force parallel to z at any point of this plane must to the first degree of approximation be equal to 



/ X the accelerative force at the point of intersection with the axis, and that the corresponding 



velocities must be in the same proportion. Hence, — being the velocity at the point of the axis 



dz 



of z, we shall have w = f -^ . It follows that (p = - a'J(pdt, and tliat c' = 0. 



For reasons which will appear hereafter I shall also suppose that c = 0, and c' = 0. Thus we 



shall have, 



, df . df d<p 



11 = —^ ; V = d) — ; w =f~r- • 

 ^ d.v ^ dy -^ dz 



It is to be remarked that these equations are the more exact, the smaller the ratios of u and !' to tv. 

 From the foregoing reasoning it follows that 



df df d(h 



ud.v + vdy + lodz = d) ^- dx + (h -^ dy + f—^dz = d .fd>. 

 ^ dm dy ' dz 



Hence iidw + vdy + wdx is an exact differential; and it is well known that in a case of fluid 

 motion in which the first power of the velocity is alone retained, this condition must be fulfilled. 

 The assumed law of the distribution of the density consequently satisfies a necessary analytical 

 condition, and on this principle is justified. It follows also that dxj/ = d .f(p, and by integration 

 that \p -ftp + a function of < = 0. Thus the equation of the surface normal to the directions 

 of motion is to a certain extent determined, and we may now proceed to obtain an expression 



for — + — ; . 

 R R' 



The known general expression for — h — ; is, 



,/rf'>// cfy// rf-\/,\ ^d^p■ d^,- d'^'. (/-«// d^// dr-j^ djr 



/dsj/^' dyj,^ d^j,'\-i ^yd.v' dy' dz' I \ da^ df '^ d«-l dx' ' dx' dy- ' dy' j^ 



[dw' dy' dz-j " 1 rf'v// d^' d'f rfx/, d^ ^ d'xj, d\f, d^l, ^ d'\j, d^ dv/, ( ' 



\ dz' dz" d.rdy' dx' dy ~ dwdz dx' dz dydx dy d« 



