486 Mr. A. M c Aulay on Quaternions as a 



II. Curvilinear Coordinates. 

 Let <f) be the stress-function, so that the equation of motion 

 of an elastic solid is 



0A + DF = De, (17) 



where D is the density, F is the external force per unit mass, 

 e is the (small) displacement of a point, and </>A is defined by 

 the equation 



^f+f + ¥ ^ , 



Required to deduce from equation (17) Lame's transforma- 

 tion into curvilinear coordinates. Changing the p, S, T, U, 

 d/dsi, d/d«2> "dfdsst, of equations (43), § 237 of Ibbetson's 

 ' Elasticity/ into D, L, M ; N, D^, I),,, D^ respectively, those 

 equations become 



DsP + D„lsr + D^M + D(B -u) 



= (P-Q)^+(P~R)^ + N(2.^ + ^)- h M(2.^ + ^ ) (if 



and two similar equations. The notation is as follows : — 



= constant, rj = constant, £= constant, . . (20) 



are three families of surfaces cutting everywhere orthogonally. 

 jut , jut,. * are defined as the curvatures of the normal sections 

 of the f surface through the tangents to the f rj and f J curves 

 respectively. If I, J, K be unit vectors normal to the three 

 surfaces f , w, f respectively, then P, Q, R, L, M, N, 5, H, Z, 

 m, v, iv, are defined by the equations 



£1 = PI +NJ +MK, &c, &c. ... (21) 



F=5I + HJ + ZK (22) 



e = ul+vJ +ivK (23) 



Lastly, Df, D Yl , D$ denote, not differentiations with regard 

 to f , 7), £, but differentiations jo^r unit length in the directions 

 of I, J, K respectively. So much for the rather formidable 



* It will be noticed that while giving the same definition as Ibbetson 

 (« Elasticity,' § 232) of jut^ jut^ equation (19) is not the same with 

 reference to these symbols as his equation (43) § 237. This is because 

 his definition is inconsistent with the meanings he assigns to the symbols. 

 To obtain those meanings he ought to give the inconvenient definition that 

 jut , jus , are the curvatures of the normal sections of £ through the 

 tangents to |£, £77 respectively. This is simply illustrated by equation 

 (66) § 243, where with 6 for colatitude and a for longitude he asserts 

 that 0' 57 a , = — cot $/?', whereas with his definitions, which I adopt, this 

 should clearly be e ^ r = —cot 6/r. 



