\ N 



T A N 



of the inosi . 

 curve y - . 



; HVATUBK, to the 



, <t>'< \+4*> \ .[ i+^ ]_ 



-"+ ?>:) +{*-+"$" 7* 



/ - 2\1 



Vl+0'u J 



Tins circle cu' e, generally speaking : it not, a- 



\amplc. at tin : an ellipse, il u evidence 



that the circle lias a 



'Jin- until nl llir circle of curvature is :i point on the 

 normal. being that :it which the normal touches tl 



Illtf. [IrsVUU'TK AM) KVOLITE.] 



only i> tin 1 ti-nn tangent most generally applied to 

 the closest straight line only, hut frequently only to that 



'.MI of the straight line which falls between [fa point 

 i'i contact and the axis of .r. Again, the nonnal is a 



_-lit line perpendicular to the tangent, drawn through 

 the point of eontaet : hut this term also is frequently ap- 

 plied only to that portion which falls between the jniint of 

 contact and the axis of .7-. It is with reference to this 

 limitation that the terms subtangent and subnormal are to 

 lie understood : the first meaning the distance from the 

 loot of the tangent to the foot of the orilinate: the 

 that liom the foot of the ordinatr to that of tin- nonnal. 

 The funuula for the subtangctit is <fn-^-<fi'.. : that for the 

 subnormal ^w/X(^'o. 



/} be the angle made by the tangent with the a\;> 



: usually the angle made In I hat part of the '; 

 which has positive ordinates with the ]iositi\e side of the 



.if .r. Then /3, at the point whose al>- . is de- 



termined by the equation 



tan /3 =^y ; and subtangent = y-r-, subnormal = tj-~-. 



If we take the more general mode of measurement pro- 

 posed in Si^\, this equation remains equally t me. Now. 

 Keeping strictly to that mode, let ft be the angle made by 

 the tangent with the axis of j; t* the angle made by the 

 radius vector r with the axis of./-, and p that made by the 

 nt with the radius vector. It will be lound. then', t hut 

 in all cases 



tie 



Unless the mode of attributing signs be carefully at- 

 tended in. these last equations, though alwajs considered 

 a.s universally true. ;ire not so in reality. 



\\'e now come to the consideration of a surface. The 

 mode of defining contact of a given order resemble* that 

 adopted with reference to a curve. Thus if :~<f> 

 and i=C i. if be the equations of two surfaces coincid- 

 ing when ./ = </, y = b. so that <(o, 6) then if 



the point be taken at which .; <i+A. tf=l> + li. \\\. 

 tact i I the two surfaces is of the n\\\ order, when the 

 deflection 



<t> (a+h, 6 + k)- + (a+t,, b+ k) 



being developed iu powers of // and k by Taylor's Theo- 

 rem. shows M.' tmiis lower .than those of the form 



'A + ... + MA". This is tantamount to the 

 following: two surfaces have ..i'the /dh onler 



1'lau.' whatever drawn through the point ol 

 't cuts Hi. in two cunes which have a con- 



tact of the /<th or a higher order. 



ha.-, at c\ci\ pom! a plane which has a 

 complete ..rder. If z - <fj 



.i.y.z be the co-ordinates of the point of ei 

 . i/. i tin. si of anj |),,int in the tangent plane, then the 

 equation of I he tangent plane ia 



>/: ,1: 



--'= +rtV ( '~ 



But if the equation be gi\cu in the form <jj ' !/ - 

 it i> 



l<t> it<t> il<t> 



-' + S "-*" + ^ ({ -* )=o - 



In (he i jimlioiiii of the normal, a line 



drawn through the point of contact perpendicular to the 

 tangent, are 



-* + -*) A t- 



In the latter ca.-,c. they are 



t-j _ - y 



= 



I. not cut 



t 



The tangent ( 

 in a sphere : -J. 



whole line. ILs III the 



as in the ease ui :tn h\ |-cil>oloii1 nun! 



'()! about 1he i the lig'u: 



The ciilerion i.f distinction belv. 

 casts depends on the \alu. 



ft #s #: \- 



nl the point of contact. Imagine a plane to p. 

 the normal, cutting the surface in the c'.i ,| t|,e 



tangent plane in the straight line : . while tin- 



plane revolves about the normal. , 1. is a; 



1. Let I be positive. Then I. h. 

 contact of the iirst onler with i(' . the - 

 passes tlnoiiLrh the tangent phine. and we ! 

 such contact as is seen at any point of a 

 ellipsoid. 



2. Let U=0. Then 1. has never more than a cot 

 of the tirst order with s('. except whi: 



one position, in which there is a con!:. 



order. If U=0at the point ol coir 



to take value ill all adjacent points, nothing' u: 



appear than in the la^t ca.se. except that in on. 



direction from the point of contact, and in 



surface would seem to grow nearer to t. 



than in any others. Hut if L T =l) at all ) 



face, this approach to the tangent plane in one piu-t 



direction becomes more- maiked: for the 



that plane in a straight line. Ih 



plane meets the surface in u straight l;ii 



tended both ways: and the plane is tangi 



at every jwint of that straight line. Such - 



those in which I! is a] ways =0. are developable, or can he 



unrolled without any overlapping, rumpling, i 



- and cylinder- m. if I' r 



throughout me whol but throughoi,' 



ticular line upon it, that line will be a plane 

 its plane will be tangent to th> mt in 



which it meets the surface. 



:t. J.et I he negatne. Then I. ha- i 

 a contact of the tirst order with ' 

 t'erent positions, in both of which I 

 higher order. Draw lines marking out th. - 



.and conse<|Ueutly dividing the tangent 

 lour parts, with four angles lournl the p, 

 In one ]>air of the . 



one side of the tangent plane, and in the oil' 

 other. 



lia, as the plane which revolves mum! (he normal 

 lakes its di tl'e.rcnt positions, tli 

 ( ' changes. The tvvo p, 

 which the 



inlo the ma 



jeet. hut shall . jxipulHr ill 



tion of this rem:.:l,ab!e point. 



-II. unbroken, to 1 with 



cither veil. i.'ist. The descent will In- equallj 



nipid in all directions, or the curvature at the h' 



.if all the \crtieal svctjons will be the same, lint 

 '"II to be so placed that some 



, dilate between the two ve- 1 

 The descent will not then be equn' 



is, or the curvatures of the > iions will 



not be the same. The direction of 

 will be at right angles to that ol 



.I plane has here a contact ol' the Iirst ,| th, 

 kinds above mentioned. If tl contnet of the 



I kind, all the eirciin \e'pt 



that the direction of least rapii! 



Uvelyspeaking.no descent at all nt the t It 



we take a cylinder, or other developable wirlace, and 



