...equations (12) and (13) may be written _ _ _ 2vdydz ~ dydt ' dx d* ,. <ty , . (17) 2 ir dx dz dxdt ~ dy where the values of the expressions are those corresponding...with respect to y, and add the results, we obtain — + — £ = 0. (18) die2 ay2 The only value of \l/ which satisfies this equation, and is finite...

...surface of the current-sheet. Hence, equations (12) and (13) may be written . dx Zif dadi dxdt^- ' ' dy where the values of the expressions are those corresponding...differentiate the first of these equations with respect to a\ and the second with respect to y, and add the results, we obtain -ry + y T =0. (18) The only value...

...— c^\ — 2#2y. (l& dx dv Hence -£ + ~ = 4//- (6) Also by differentiating equations (5), the first with respect to x and the second with respect to y, and substituting for — and -J^-, whenever they occur, their values given in (5), we have Hence (4) becomes,...

...(67) will give ' .(86). Moreover the equations (69) will become ^ 2 * Differentiate the first of (87) with respect to x, and the second with respect to y, and add ; thus .-., fd'aj, d8w0\ dX, dY0 Finally the equation (27) will give , t' /d'à,, eP»A »• / AC...

...positive surface of the current-sheet. Hence, equations (12) and (13) may be written <r d2P d2 d\)r where the values of the expressions are those corresponding...with respect to y, and add the results, we obtain The only value of \jr which satisfies this equation, and is finite and continuous at every point of...

...positive surface of the current-sheet. Hence, equations (12) and (13) may be written Zvdydz -._*_ > *+ where the values of the expressions are those corresponding...with respect to y, and add the results, we obtain jTj-5 ' dx* dy2 The only value of -fy which satisfies this equation, and is finite and continuous at...

...dw m — 1 ' dx dy m 3A dw _ 1 >r I /* ^"™ " (13) m — 1 dy dx Differentiating the first of these with respect to x and the second with respect to y, and adding the results we get, after dividing by m/(m — 1), or V2A =0 (14) r- d* •where V2 stands for...

...of equilibrium are 8crx 8rxy | "5 , ~5 — — «i ox 8y dcry 8rxy ~o ,~s — — VJ. ay ox (1.16) If we differentiate the first of these equations with...respect to x and the second with respect to y and add, we get a^+8^= ~28xOy- (L63) Solving (1.62) and (1.63) to eliminate the derivatives of TXV and re-arranging,...

...region D these solutions can be many-valued. Proof. Differentiating the first equation of system (16) with respect to x and the second with respect to y, and adding, we obtain - [(ow), + (Pa),] [/ = (aJbV), + ($bV)y (29) We investigate the question of the solvability...

...resultant deflection defined by Eq. (2.64) in the component theory. Differentiating the first of Eqs. (3.6) with respect to x and the second with respect to y, and substituting the corresponding expressions from Eqs. (3.4), we obtain "|-*/2 da)x 12 {+ ' Su , 12 f+*/2r...