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 

 zinc 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. 



University of Virginia, Oct. 9th, 1833, 



Art. IV. — Motion of a System of Bodies j 

 by Prof. Theodore Strong. 



Continued from p. 46, Vol. xxiv, 

 ANALYTICAL FORMULAE. 



It will here be convenient to give the investigations of some ana- 

 lytical formulae 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,s, 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 w, 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 (T) ; also let a, b, c, denote the co- 

 sines of the angles which (1) makes with the axes, x, y, z, severally, 

 and a', b', &, the corresponding cosines for (!'); then a, b, c, are the 

 orthographic projections of (1) on the parallels to the axes x,y,z, 

 severally, and a', b', d are the projections of (1') on the same 

 lines ; (for the orthographic projection of one straight line on anoth- 

 er, equals the line to be 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=l ;) (p=the 

 angle made by the given lines ; then (1) projected on (l'')=cos. (p, 

 but the projection of (1) on (1') is evidently equal to the sum of the 

 projections of a, &, c, on (T) ; now the projection of a on (1^), -^aa', 

 that of 6, =66', and that of c, =€</,.'. cos. 9=«a'+66'-i-cc^ («) ; 



Vol. XXV.— No. 2. 36 



