93 
always of the sum of two parts — one immediately integrable, 
the other integrable by a factor. 
Let dF be the integrable part. Then 
is integrable by a factor if 
VdQ 
ent\ 
dF/'dR 
dP> 
'dP 
dQ\ 
\ dz " 
’ d v) 
dy\dx 
dz ) 
l + &( 
\dy ~ 
“ dz) 
-<i-£M£-SM2 
clF\ 
dy) 
But this is an ordinary linear partial differential equation, 
from which F can be found as a function of x y and 0 by 
Lagrange’s method, and, therefore, a function F can always 
be found such that P dx + Q d,y + Fdz - dF is integrable by a 
factor. 
5. Returning to the form of a - v P + £ we see that 
D 4 <t = D*vP + D^. 
- -S(vP + i;)v^vP + Vp£ + S£ v p by 1 
= (~~ t - S^Pv) VP + Vp4 + Sfvl 
= (|- SvPvjvP+lv?.* 
Where if K did not exist D,<r would be ^-SvPvjvP 
showing that the accompanying irrotational motion is one 
capable of existing by itself, while the essentially rota- 
tional portion is not generally capable of so doing; and that 
IV is not D t v P(£ — 0) + D*£( v P = 0). 
6. If we find the angle between the axis of rotation at 
any point and at a distance x along the filament, we may 
deduce a measure of the curvature of the filament, and by 
taking a corresponding length on an axis perpendicular to 
the osculating plane we may form a vector of curvature at 
the point. 
# Notes on Quaternion transformations. 
