...orthogonal, we must have identically UU, + FF, + WW, = o * This demonstration of Dupin's theorem was given by Mr. Thomson, of St. Peter's College, in the Cambridge...to zero, in the result, and making use of equations (6), we have the brackets denoting that, in the quantities enclosed, x, y, z, are equated to zero....

...VtV + w^W = 0 ............... (c). Differentiating the first of these equations with respect to ж, the second with respect to y, and the third with respect to г, putting ж, у, г, each equal to zero, in the result, and making use of the equations (6), we...

...MV v = 2 Y. dt ay * ' dw dU ,. „ _____МУМ, = 2£ Differentiating the first with respect to ж, the second with respect to y, and the third with respect to г, adding and using the equation of continuity, we have u fdZ dY dZ\ - _++ therefore . dY dZ p-=CT*...

...reduces to a linear function of x, y, z. In fact suppose we differentiate the first of equations (6) with respect to x, the second with respect to y, and the third with respect to z ; we shall have A , cPf *? , *f _n Jf , A _n fn -. — 5 h 7 — j— = «. -j — ; Tj ,- — U, j...

...dy \Ve must assume that /, has no longer the above value ; but by differentiating the first of these with respect to x, the second with respect to y, and the third with respect to :, we get — dx ,ix dy + Q ,/z d fdù dt\dx df dy du>\ ÍH0^ + dy dm\ „(P2 + QJ + R2 + o2 + fij...

...2M) . 36/3* + 2/*(3Ax/3y - 3Vav) + p^ = o' If the first equation is differentiated with respect to z, the second with respect to y, and the third with respect to z, we have by addition when p is constant, (A + 2/*)y26 + P(dXpx + dY/dil + dZ/dz) = 0. If X, Y, and Z...

...equations (c) become (e) (A + 2M) . 38/3; + 2/ If the first equation is differentiated with respect to r, the second with respect to y, and the third with respect to c, we have by addition when p is constant, (X + 2/i)v-9 + p(dXfdx + 3r/3y + VZj'dz) = 0. If X, Y, and...

...has no longer the above value, as X, Y, Z do not vanish ; but by differentiating the first of these with respect to x, the second with respect to y, and the third with respect to z, we get dX dY dZ d Ida do dw\ dx dy dz dt \dx dy dz J p de r\de ^ dc \ f ^ dm _ dm dm\ dx dy dzj \ dx...

...with respect to x ; in (7), according to (1), (2), (3), and (5), the first integration is to be made with respect to x, the second with respect to y, and the third with respect to x. That is, the first integration sign on the right is taken with the first differential on the left,...

...in (7), according to (1), (2), (3), and (5), the first integration is to be made with respect to z, the second with respect to y, and the third with respect to x. That is, the first integration sign on the right is taken with the first differential on the left,...