19(5 Mr. F. L. Hitchcock on 



Then by taking a unit along V?? 



y =UVP = (2/oSp^/o + ^TVy) (TVy Sp<£/> - 4a 2 S>fy>) -*, 



T 2 <£|0 being here the same as Spc£p. In differentiating again 

 it will be well to put, for brevity, T <2 q = t 2 and $p<j)p = s' 2 , so 

 that we have T 2 p = t 2 -a 2 . The result is 



Ov = xdp = {dp(2tV —8a 2 /) + ^fpSpdp -f (ppSdp^p^crs-t 2 — f) 

 + (frdpisH* — IctsH 2 ) + 2s 2 t*p8dp(f)p + ScfstypSpdp} {s 2 ^ - 4aV} - 1 



This linear and vector function contains six vector terms, of 

 which all but the last two are self-conjugate, and therefore 

 contribute nothing toward Wk The last two terms give 



YVv = (2s 2 * 4 - $aY)Yp(j>p («V - 4a 2 /) - * 

 = 2Y P (f)p(s 2 t i -±a 2 s i )- i . 



If this last expression be substituted for dp in ydp above, all 

 the terms vanish except the first and the fourth, giving 



x YVv = (±s*V P <l>p + 2t 2 cf>Y p$p) (V¥ - 4«V) - l . 



But bv an elementary transformation (Kelland and Tart's 

 i Introduction to Quaternions/ p. 190, r), since <p is self- 

 conjugate, we have 



<j>Y P (f)p = — lYpfyp - Ypcj> 2 p, 

 and also 4> 2 P = ""^Pj whence 



x YS7v = 2Yp(l)p(2s 2 -i 2 )(i-t i -±a 2 s 2 )-\ 



which is a scalar multiple of Y\7v. Thus Ypcj>p is a vector 

 parallel to r) and 



S V VVp<f>p = S(fr -jw)v(iy-j*) = 8B* fy -/,■) = 0. 



It is clear also that (19^/) reduces to 



dm ' 



a general property of surfaces of revolution,, provided the 

 axis is the same for all members of the family. 



The following may be taken as further illustrations : — 



1. If Sot = 0, ycr differs from y^a by a normal vector. 



2. When applied to a vector in the tangent plane the 

 operator )([}'%{v( )}] or (^Vi/) 2 is equivalent to a scalar. 



