Motion of a System of Bodies. ~ 281 
rents exist in mines. Without denying the fact, it is obvious that 
those obtained, were the result, solely, of the instrument employed. 
For if we connect a disc of zinc with one wire of the galvanometer, 
while the other wire touches the moist iron ore, and then bring the 
zine into contact with the fluid upon the ore, a current will result, 
which proceeds from the zinc alone, in consequence of its oxidation. 
I am inclined to the opinion, therefore, that these experiments, upon 
the strata of mines, admit of this explanation in all the. cases hitherto 
noticed. ye po 
University of Virginia, Oct. 9th, 1833. 
Art. IV.—Motion of a System of Bodies ; 
_ by Prof. 'Turopore Srrone. 
Continued from p. 46, Vol. xxiv. 
ANALYTICAL FORMULAE. 
It will here be convenient to give the investigations of some ana- 
lytical formule which will be wanted in the course of this paper. . 
(1.) To find an expression for the cosine of the angle made by 
any two straight lines, in terms of the angles which they make with 
three rectangular axes, x,y,z, drawn through any given point. 
If the lines intersect ; through their point of intersection, draw three 
straight lines parallel to x,y,z, then evidently the given lines will 
make angles with x, y, z, which are equal to those which they make 
with their parallels respectively. ‘Take on each of the given lines a dis- 
tance (from the angular point,) equal to unity=the radius of the 
trigonometrical tables; let one of these distances (for distinction,) 
be denoted by (1), the other by (1’) ; also let a, 6, ¢, denote the co- 
sines of the angles which (1) makes with the axes, 2, y, z, severally, 
and a’, b’, e’, the corresponding cosines for (1’); then a, 6, c, are the 
ssiinacephiic projections of (1) on the parallels to the axes a, y, 2, 
severally, and a’, b’, c’ are the projections of (1’) on the same 
lines ; (for the orthographic projection of one straight line on anoth- 
er, equals the line tobe projected, multiplied by the cosine of the 
angle which the lines make with each other.) Let P=3.14159 etc. 
(=the semicircumference of a circle whose radius=1;) 9=the 
angle made by the given lines ; then (1) projected on (1’)=cos. 9, 
but the projection of (1) on (1’) is evidently equal to the sum of the 
projections of a, b,c, on (1'); now the prgecton of a on (1’), =aa’, 
that of 6,60’, and that of c,=cc’, .’. cos. p=aa’+-bb’+ce’, (a); 
Vor. XXV.—No. 2 36 
