By Alfred Barnard Basset

Initially released in 1910. This quantity from the Cornell college Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 structure by way of Kirtas applied sciences. All titles scanned hide to hide and pages may possibly comprise marks notations and different marginalia found in the unique quantity.

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**Extra resources for A Treatise on the Geometry of Surfaces**

**Example text**

Conversely any replica of A is of the form 3, where B is a replica of A. Proof. For simplicity, we prove this lemma in the case =! •A ( A )• w If e represent A as in (4). +*< A,, V ^=(V) where The linear space on which A operates is ® =TO+ 2' cteristic roots of A are A,, \ x* and and the chara- ri 2 r * = - \ , - - - - V + X X J , + --+ J«= 1-1 (- ft, - • • • - tf,e -f- > s v + - + r )x v lt where (,, *•-, iV, }\, <•• , / . are chosen arbitrarily from 1, 2, A, If « ^ _ x _ . . - x . + X + . .

Matsushima [1). 3) Our /-algebraicily is the same as the algebraicity i n C h e v a l l y and T u a n ' s sense. 4) M . Goto [2]. 5) L . Maurer [1], S. L i e and F . E n g e l [1] p. 800-307. 6) Y . Matsushima [1]. 7) I owe this remark lo M . C o l o . S) At and B> are commulalive (/=1, 9) See. M. Goto proved also our Theorem 3 and 4 i n a different , k; r = l , way. ,_/). Cf, the proof of L e m m a 1 below. 10) a=\ i,-\- Y . Matsushima [1]. A s usual 0>K is a L i e algebra over + Ar*ri where J j t A ' a n d e,, y 11) E .

D X=[AX] for Xt2. is represented by ^-matrix or j-matrix according as A is //-matrix matrix. Then Thus 1 inner D or s- A A A We omit the proof. Lemma 8. QI («, P). Then there exists an //-matrix A and an s-matrix A in gl (;/, P) such that [A°A ] = 0 and ^ ^ " - t - ^ . Moreover they are replicas of A ^. Proof, Let R be the splitting field of the characteristic equation of A . r-matrix respectively such that [A A')=0. Now let a be any substitution of the Galois group of K/P. A'~\ = 0, Hence, by the uniqueness of such decomposition, we get oA»=A\