A. \AI.Y IK <,!:< >Mi:ii;Y 



[Cii. \ I.. 



The subtangent is tin- length 7M7. where M is the foot of 

 the nrdin.it.' <!' the point i.f tani^-nex l\ ; and tin- subnormal 

 is the corresponding length MN. As thus taken, tin- sub- 

 tangent and the subnormal are of the same sign ; ordinarily, 

 however, one is concerned merely with their absolute values, 

 irrespective of the algebraic sign. The subtangent is the 

 projection of the tangent length on the z-axis, and the suit- 

 normal is the like projection of the normal length. 



87. Tangent and normal lengths, subtangent and subnor- 

 mal, for the circle. The definitions given in the pm-rdin^ 

 article furnish a direct method for finding the tangent and 

 normal lengths, as well as the subtangent and subnormal. 

 for a circle. Eg., to find these values for the circle 



2?+y 2 = 25, and correspond- 

 ing to the point of contaet 

 (3, 4), proceed thus : 



The equation of the tan- 

 T x gent PjZMs (Art. 84) 



Fio.68. 



hence the re-intercept of t his 

 tangent, i.e., OT, =^; 



therefore the subtangent TM, which equals OM ' OT, is 



3 ^, i.e., - 5J. The tangent length 



TP l = 



+ 4 2 



To find the normal length, and the subnormal, first \\iito 

 the equation of the nonmal at the point (3, 4); it is (Art. 

 85) 4 a: 3y = 0. Hence its ^-intercept is zero, and tin- 

 subnormal, M in this case, is 3 ; the normal length J\ O 

 is 5. 



