266 
MR. MACQUORN RANKINE ON AXES OP 
Invariants of the second order in Umbrce are real quantities of the first order, viz. — 
(a^) +( 7 ^) +2(|37)+2(705)+2(o5|3) = (^i)^ (the cubic elasticity) 
m+(7C‘)+w- w- (f*’)- (■’’) =(« 
(e) +(,3>) +( 7 ») + 2 (X’) +2W + 2 (.*) =(0,)=-2(4) (10.) 
The equation of a Tasinomic Sp]iere\8 formed by multiplying aTasinomie Invariant by 
or any power of that quantity, and equating the result to a constant. 
6 . Of Two Tasinomic Ellipsoids, and their Axes, Orthotatic and Heterotatic. 
The equations of two Ellipsoids with tasinomic coefficients are derived from that 
of the Umbral Ellipsoid (8 a.), in one case, by multiplying each term by the Umbral 
Invariant (^J, and in the other, by substituting for each Umbra in the function {<p), 
the contravariant component of the Inverse to the Umbral Matrix (9.). The results 
are as follows : — 
Orthotatic Ellipsoid. 
(^ 1 ) X (?>) = { (a"*) + (cc^) + ( 7 a) } { ((3^) + ((3y) + ( 705 ) 
+ { ( 7 ^) + (7“) + 
+ 2 { ( a7) + (^7) + (yX) + 2 { (a^) + (^^) + ( 7 ^) } 20 ? + 2 { (ar) + ((3p) -\-{yv)]xy=\. (11.) 
« 
Heterotatic Ellipsoid. 
{ (M — (7«) - (f) }f+ { W — f ) } 
-i-2{(fiv) — (ccX)}yz+2{(vX) — ((3f/.)}zr-l-2{(Xfo)—(y!^)}ry=l. . . (12.) 
The three Orthotatic Axes are three rectangular directions for which the following 
sums of Plagiotatic Coefficients are null : — 
(aX) + (^X) + (77) = 0 ; (a^) -f (/3 /a) + ( 7 //,) = 0 ; (av) + (/3j/) + (7v)=0. . (13.) 
It w^as proved by Mr. Haughton, in a paper published in the Transaetions of the 
Royal Irish Academy, vol. iii. part 2, that there are three rectangular directions 
having this property in a solid whose elasticity arises solely from the mutual actions 
of physical points, and which has but fifteen independent coefficients of elasticity. 
The present investigation shows that there are three such axes at each point of 
every solid, independently of all hypothesis. The physical meaning of this result is 
expressed by the following 
Theorem as to Orthotatic Axes. 
At each point of an elastic solid, there is one position in which a cubical molecule may 
be cut out, such, that a uniform dilatation or condensation of that molecule by equal 
elongations or equal compressions of its three dimensions, shall produce no tangential 
stress on the faces of the molecule. 
