The Rev. J. H. Jellett on the Properties oflnextensible Surfaces. 349 



Substituting for Sp, hq, and reducing the resulting expressions by means of 

 equations (D), (E), (F), we find ultimately, 



/ , , „ , d-cz fdlx (lcii\ „ / dlx dcy\ 



,. , ,.d^lz J dlx dh/\ „/ dlx dcy\ 



From the two equations (G) it is easy to verify that the element of the su- 

 perficial area remains constant ; for if we multiply the first of these equations 

 by p, and the second by q, and add them, we find, recollecting the second of 

 equations (B), 



pip + qlq =(!+/ + r/) U 1^ + y ^ j ; 



or from tlie first and third of equations (B), 



^,;, + ,,, + (l + ,= + ,=)(|%|^) = 0. (I) 



which is obviously equivalent to 



I -/(l+p' + y') dxdy = 0. 



Again, multiplying the first of equations (H) by <, the second by 2^, and the 



third by r, and subtracting the second product from the sum of the other two, 



we have 



tlr + rlt - ^sls = l{rt- s^) 



Hence, and from equation (I), it is easy to see that 



^Crt-s") _ (pip + qlq) _ 

 rt-s' 1 + f + q' ' 



which is plainly equivalent to 



2z2 



