238 THIRD REPORT— 1833. 



would properly be interpreted to mean that under no cir- 

 cumstances whatever, whether in the plane of x ?/ or in the 

 plane at right angles to it, in which the hypei'bolic portions* of 

 curves expressed by those equations are included, would a point 

 of intersection or a simultaneous value of x and y exist : or in 

 other words, the sign or symbol oo would in this case mean 



that such intersection was impossible. If we supposed — = — ^ 



and also b = i',or the ellipses to be coincident in all their parts, 



then we should find a:- = -y- and y = -^, indicating that their 



values were indeterminate, in as much as every part in the iden- 

 tical curves would be also a point of intersection, and would fur- 

 nish therefore simultaneous values. If we should suppose b 



greater than b', a greater than a\ and -j- not equal to -jy , then 



we should find 



s = a and i/ = /3 \/— I, 

 or X = a. a/ — I and y = /3, 



according as -j- is less or greater than -y?. In this case, one 



ellipse entirely includes the other, but the hyperbolic portions 

 at right angles to their planes, which are in the direction of 

 the major axis in one case and in that of the minor axis in the 

 other, will intersect each other at points whose coordinates are 

 the values of x and y above given : it would appear, therefore, 

 that the impossible intersection of the curves would be indi- 

 cated by the sign or symbol oo alone, and not by \/ —I. 



The preceding example is full of instruction with respect to 

 the interpretation of the signs of algebra, when viewed in con- 

 nexion with the specific values and representations of the sym- 

 bols ; and there are few problems in the application of algebra 

 to the theory of curve lines which would not furnish the mate- 

 rials for similar conclusions respecting them : but it is chiefly 

 with reference to the connexion of those signs with changes in 

 the nature of quantities, and in the form and constitution of ex- 

 pressions, that their interpretations will require the most care- 

 ful study and examination. We shall proceed to notice a few of 

 such cases. 



X' y^ I— 



* If in the equation — -|- -p- = 1, we suppose y replaced by y v — 1, 



and the line which it represents when not affected by V — 1 to be moved 

 through 90° at right angles to the plane of x y, we shall find an hyperbola 

 included in the equation of the ellipse. 



