I 



OF SYMBOLICAL GEOMETRY AND MECHANICS. 503 



To deduce from this expression the common equation of the normal plane. 

 Let x^y^z^ be any point in the normal plane; then 



„ ldu\ 

 ,Ta + 2/, p + x^y = u + md\ — \ 



Hence we have 



.dx dy .dss 



ds •'' ^ ds ' ds 



- dw , dy dz m r, Udsf] 



and .-. (.r -*■)_+(« _ „) -^ + (« _ sr) — = — ^ <f^> = 0, 

 ^ ' ds ^■'' "' ds ^ ' ' ds 2 \{dsf) 



which is the common equation. 



28. It is however much more convenient to use the symbol D in expressing perpendicularity. 

 Dv.du denotes a line of any length perpendicular to du, supposing v to be any arbitrary line 

 symbol. Hence the formula of the normal plane is 



u + Dv . du. 



29. The formula of the normal perpendicular to the osculating plane is 



ti + niDd'u . du, 

 because du and d?u both respresent lines lying in the osculating plane. 



30. We shall now give a few examples of the application of this method to surfaces and to 

 some common geometrical problems. 



If u be the formula of a surface, the formula of the tangent plane at the point m, is 



. u + mdu, 

 m being a numerical variable parameter. 



For du represents the elementary line joining any two contiguous points of the surface, and 

 therefore mdu represents a line of any length touching the surface at the point u. 



.31 . The formula of any normal plane (i. e. any plane containing the normal at the point zi) is 

 evidently 



u + Dv . du, 

 I! being any arbitrary line symbol. 



u, being the formula of a surface, must involve two variable parameters: let them be m and n 

 (both numerical), and let d,„u and d„u represent the respective partial differential coefficients of m 

 with respect to m and n : then the formula of the normal at tlie point m, is 



u + pDd,„u. d„u, 



p being a numerical variable parameter. 



.32. The formula of a plane containing tlic three points u, u, u' , is 



u ■¥ m (u - u) + n (u" - m), 



or what is the same thing, 



mu + m'u + m"u". 

 Where m, in', m" arc numerical parameters subject to the condition m + m' + m" = 1. 

 Vol,. VIII. Taut IV. 3 T 



