464 Mr MAXWELL, ON THE 



We may simplify these equations by putting p for the specific curvature of the surface, 



and q for the ratio "—, which is the only quantity altered by bending. 

 P 



We have then 



P = , • ; , » and q = ^ , 

 pp sin <p p 



whence p 2 = q — — — , p n ■ 



p sin 2 <p ' " q p sin 2 (p ' 



and the equations become 



-(lo gg )=-log^_l j +7 _ cot0+- - gTn - 



4> ? 



d _ i d , / dsl 2 \ 2 A' , 2 ds' 1 1 



^' G0g?) = -dV l0g ^^|)-r dl? C0t ^-/d7sin^i- 



In this way we may reduce the problem of bending a surface to the consideration of one 

 variable q, by means of the lines of bending. 



16. To obtain the condition of Instantaneous lines of bending. 



We have now obtained the values of the differential coefficients of q with respect to each 

 of the variables u, u. 

 From the equation 



we might find an equation which would give certain conditions of lines of bending. These 

 conditions however would be equivalent to those which we have already assumed when we drew 

 the systems of lines so as to be conjugate to each other. 



To find the true conditions of bending we must suppose the form of the surface to vary 

 continuously, so as to depend on some variable t which we may call the time. 



Of the different quantities which enter into our equations, none are changed by the 

 operation of bending except q, so that in differentiating with respect to t all the rest may be 

 considered constant, q being the only variable. 



Differentiating the equations of last article with respect to t, we obtain 



' i (l0g9) %£^ 9 ^ (1 ° g9) ' 

 d 2 2d*' 1 Id 



du'dt r du sin <p q dt 



_ d 3 _ „ f d /2 ds 1 \ 2 ds 1 d . . 1 d 



Whence Qagq) «{— j (- —- -— -) + - — -— - —-, (log?)} q ~r. (log q) 



dududt [du \r du sind)/ r du sin <p du ] at 



2 ds 1 d 



+ rdu ^n~0 q dVdi ( ° g g) ' 





