MATHEMATICAL SECTION. 93 
In his own investigation Mr. Exttortr 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 @. 7) (cos 0’ + sin 0’ cosw.i+sin@ sinw.7)= 
cos 9 cos 0’ — sin 6 sin & cos w 
+ (sin @ cos & + cos 0 sin & cos w) t 
+ cos @sin # sinw.j 
+ sin @sin 6 sin w .77. 
In this expression 0, 6’ and w have the same signification as stated 
above, and 2, 7 and 7% (or its equivalent &) 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, ete. fol. New York, 1887, Au- 
gust 27; vol. 10, no. 9, p. 116. Also separately printed. ] 
Mr. Hit, following Maxwell, gave some of the principal steps 
in the process which leads to Ampére’s result, indicating that that 
process differs from Mr. Exuiort’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. Doo.irrLe began the presentation of a paper on 
ASSOCIATION RATIOS. 
