80 



the locus of a point, such that tangents being drawn from it 

 to two equal intersecting circles, their rectangle may be con- 

 stant, and equal to the square of half the common chord. 



If the circles do not intersect, and if the rectangle under 

 the tangents be equal to the square of the tangent to either 

 from the middle point between the centres, the locus will give 

 the curve derived from an hyperbola, whose real axis is greater 

 than the imaginary. 



To get the curve when the imaginary axis is greater than 

 the real, we must take the locus of a point, such that lines 

 being drawn from it to meet the circumferences of two equal, 

 non-intersecting circles, and subtending right angles at their 

 centres, their rectangle may be constant, and equal to the 

 sum of the squares of the radius, and of half the distance be- 

 tween the centres. 



There exist in spherical geometry numerous properties of 

 an analogous character. 



The Rev. Charles Graves made a communication relative 

 to the new functions employed by him in the interpretation of 

 his theory of Algebraic Triplets. 



The three functions A, m, n, defined in p. 60, possess so 

 many interesting properties that they seem to deserve distinc- 

 tive appellations. The first is symmetrical with respect to its 

 two amplitudes, ^ and ^ I and its properties are in the main 

 analogous to those of the trigonometrical function cosine. On 

 the other hand, m and n are not symmetrical functions of ^ 

 and 1^ ; but either of them may be obtained from the other by 

 interchanging the two amplitudes; and they correspond in many 

 respects to the trigonometrical sine. Mr. Graves proposes 

 then to call a the cotresine, and m and n the tresines of 

 and -y^ : and he designates them respectively by the symbols 

 cotr [0, x], tres [0, x], tres [x, f\. 



As in trigonometry cos i^ = cos(2i7r +0), and sin0 = 



