610 Dr. T. H. Havelock on Surfaces of 



and similarly 



we obtain 



^di^-^+^sw- • • • (18) 



But from (17) 



2 o{ox) o(oa?) d# 



Hence on S we have 



8\=ASx + B8y + CBz. 



To obtain the conditions of compatibility in the second 

 order, we take variations as before on the multiplicity 2 

 instead of on S ; the time variation St occurs in a similar 

 manner to the other variations, and introducing a fourth 

 undetermined multiplier D, we have on 2 



And the discontinuities in the second-order derivatives can 

 be expressed finally in the form 



\^x ^y J ^z ^t J J 



+ 2(g&+^«y+^S^+|f^(ASa ? + BSy + CS 5 + D&); (19) 



together with similar expressions in 



0,, ix, A', B', 0', D') and (f, v, A", B", 0", D"). 



Thus for a first-order discontinuity defined by a vector 

 (X, n, v) in equations (13) and (14}, the variations in the 

 second-order derivatives are expressible in the form 



Lb« 2 J-V + a*' ( 0) 



and similar expressions. 



