VOL. VII.] PHILOSOPHICAL TRANSACTIONS. 3Q 



For since that point is given, y and v are likewise given, as also the other quan- 

 tities expressed by consonants, consequently a becomes known. If there be 

 found any obscurity in this rule, it will be cleared up by some examples. Let 

 this equation by — ?/ = i^ be given ; in which let EB be ^?; B A, ?/; DA, v ; and 

 let a or AC be sought, so that drawing D C, it may touch the curve DQ in D. — 

 According to the rule, nothing is to be rejected out of this equation, since in 

 each of its terms either y or v is found ; and it is besides so disposed, that on 

 one side are all the members in which y is, and on the other side all those in 

 which V is found; there is therefore only to be prefixed to each member the 

 exponent of the power o£ y orv in each, and on the left side one y to be changed 

 into a, that it may be ba — 2ya = 2vv; now this equation shows the method 



of drawing a tangent to the point D, or « = ^ — = AC. And if the equa- 

 tion qq + ^j/ — yy-=.vv were given, the equation for the tangent would be 

 exactly the same with the preceding, after rejecting qq according to the rule. 

 So also from 2 by^ — y^ ■=. v"^ arises A by a — 3 yya =: 3v\ or a ^ 



■ — . And from bby -\- zyy -\- y^ ^ qvv, is obtained bba + 1 zya + 



3yya =2qvv, and a=zj^^-~^—-. Also from b^ -{- by^ -^ y' = qqvv 



+ zv^ comesSbyya— 4y^a=:2qqvv -^ 3zv^y and hence a = IV^ _ — 77"* 



In these and such like equations there can be no difficulty. Possibly there 

 may be a little in those, which have some of the terms consisting of the products 

 of ^ and i;: as yv, yyv, y^ vv, &c. Yet that difficulty is but inconsiderable: 

 for suppose we have^y^ = bvv — yvv\ nothing is to be thrown out of this equa- 

 tion, since either ?/ or?; is found in each term. But that it may be disposed ac- 

 cording to the rule, yvv must be taken twice, and be put both on the right side, 

 in which are the terms having v, and on the left side, whose members have y, 

 since yv v contains both y and v ; then we must make y^ -|- vvy= bvv — yvv. 

 And changing this equation as before into another, viz. 3yya -\-vva = 2bvv 



— 2vvv, a will be equal to J^"" ^^. For the rule is to be thus understood, 

 viz. that on the left side the power of v is not to be regarded, so that the expo- 

 nent of ?;f must not be prefixed to yvv, only that of ^ ; as on the right side, the 

 power of z/ in yvv must not be regarded, but only that of v, whose exponent is 

 to be prefixed. Thus, if it were y^ + %* :=2qqv^ — yyv ', it should hey^ + 

 by'^-\-yyv^=z 2qqv^ — yyv^; and the equation for the tangent would be oy^a 



+ 4by^a-\'2yav' = 6qqv^-3yyv^andheucea=: ^^^-o~T- 



And these examples seem to comprehend all possible variety of cases. But 

 perhaps it may be of use to apply what was explained in general to some particu- 



