of Edinburgh, Session 1881-82. 543 
sponding points of the resultant in the different states of rotation of 
the system. 
To the question, if every one of the surfaces (39) has one point 
in common with every one of the resultants, we give the answer in 
§ 9, where we show that through every point of space we are able 
to draw four straight lines meeting the hyperbola (42), and which 
satisfy moreover the conditions derived from the expression of h ' . 
We have thus the proof of one part of Minding’ s Theorem as 
announced at page 37 of Crelle’s Math. Journal , vol. xv. , namely, the 
resultant in the different, innumerable, positions which it may take, 
passes constantly through the hyperbola which is the limit of the 
infinitesimally narrow canal corresponding to h-b 2 = 0. 
The proofs of the other propositions of the theorem will be gained 
by the examination of the directions k' of the single resultant. 
§8. 
Writing again the expression (37) containing k ' , 
0 = nk! + ixili - b 2 ) +jy(h — a 2 ) 
+ k[z(h - M 2 ) + gab ] , 
we introduce into it the hypothesis 
li-b 2 — an infinitely small value. 
For the preparation of n and g as defined in (36 bis) and (37 bis ) 
we have the relations 
( (ga - bz) 2 — X 2 , 
}(as-rjbf = Y 2 , 
which become identities if we substitute into them the value of g 2 
(36 bis) and the value of 2gabz drawn from (36), and having 
regard to (32) and (38). 
As the value of y is to converge to zero, at the same time as 
h - b 2 , but independently of it, we put 
cy = w(h - b 2 ) , 
calling w an arbitrary scalar. 
We have by (43) 
gb = az- Yf , 
