MATHEMATICAL SECTION. 
93 
In his own investigation Mr. Elliott makes no assumption or 
restriction with reference to the direction of the action of the ele¬ 
ments and finds the action proportional to the following expression: 
(cos 0 sin 0 . i) (cos 0' sin 0' cos (o . i -\- sin 0' sin o) .^’) — 
cos 0 cos O' — sin 0 sin 0' cos (o 
-f (sin 0 cos 0' -{- cos 0 sin 6' cos (o) i 
+ cos 0 sin 0' sin o) . j 
+ sin 0 sin 0' sin m . ij. 
In this expression 0, 0' and o) have the same signification as stated 
above, and i,j and ij (or its equivalent k) are quadrantal versors. 
The first term of this formula represents action in the line joining 
the elements; the second term represents action in the plane of the 
connecting line and one of the elements and perpendicular to the 
connecting line; the third term represents action in a direction at 
right angles to the plane just mentioned; and the fourth term rep¬ 
resents torsion in a plane perpendicular to the connecting line. 
The actions resulting in some special cases, as when the elements 
lie in one plane, etc., were explained and discussed. 
[This paper was presented to the American Association for the Advance¬ 
ment of Science at its New York meeting, August, 1887, and appeared in 
the Electrical World; a weekly review, etc. fol. New York, 1887, Au¬ 
gust 27 ; vol. 10, no. 9, p. 116. Also separately printed.] 
Mr. Hill, following Maxwell, gave some of the principal steps 
in the process which leads to Ampere’s result, indicating that that 
process diflfers from Mr. Elliott’s in leaving out of account cer¬ 
tain couples and in assuming a certain undetermined quantity to 
be zero. 
The paper was further discussed by Mr. Harkness, the Chair¬ 
man, and others. 
Mr. Doolittle began the presentation of a paper on 
ASSOCIATION RATIOS. 
