

PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. lala 
For we should find, as their common value, the expression of their swm 
dg 
dy 
+ S.kg % 
dz° 
Sy + S.9q 
19. Also, from (30) and the other similar equations, we have the following 
series of values— | 

du ~ yl dq 
Tp ee ae 
1 du —ldq 
noe (ees 
ldu _ —ldq 
ude dz 
e (32). 
1ldu —1dq 
me oie ie 28.4 a 
1du Ng eg 
gag OF ae 
1 du = wag 
PSI es 
These give three equations of the form 
—1dq L fi.dt dw 
Vig = Oy: agen oy 5 
which enable us to make the transformation assumed in § 17 above, and which 
may be all summed up in the following—in which the omission of the V is due 
to the remark in § 6 that the tensor of g may be assumed constant— 
ING. dq —2q, dq =— V.(dpV) log. « A pss): 
This is the equation determining the quaternion which gives the position of a 
rigid body in terms of the vector-axis of instantaneous rotation. (On the Rotation 
of a Rigid Body about a Fixed Point. Trans. R.S.E. 1868-69.) 
20. But by § 6, we have 
Gi=6)-6)-—». | 
dead daa ia | ee 
a ada ao- “Uo ao ac 
Dam ery coo eee 
From these 
glances ni. du 
a ia Pe dx. 
do do do Go _ du 
dy di? — © dx dady — “ dy’ 
VOL. XXVII. PART I. 
ho 
jan 

