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LX. Thoughts on Inverse Orthogonal Matrices, simultaneous Sign- 

 successions, and Tessellated Pavements in two or more colours, 

 with applications to Newton's Rule, Ornamental Tile-work, and 

 the Theory of Numbers. By J. J. Sylvester*. 



Part I. — Matrices and Sign-successions. 



1. A SELF-RECIPROCAL matrix may be defined as a square 

 ^*- array of elements of which each is proportional to its first 

 minor. When the condition is superadded that the sum of the 

 squares of the terms in each row or in each column, or (which 

 comes to the same) that the complete determinant shall be equal 

 to unity, it becomes strictly orthogonal ; but, by an allowable ex- 

 tension of language, any self-reciprocal matrix may be termed 

 orthogonal when the epithet of stiictness is withdrawn. The 

 general notion is that of nomographic relation between each ele- 

 ment and its first minor, i. e. the relation aA-bx + ct; -$-dx% = 

 between the corresponding terms x and f of the matrix and its 

 reciprocal. When a = and d=0, we have the case of orthogo- 

 nalism as above defined t- When b — and c = 0,so that each term 

 in either matrix is in the inverse ratio of its first minor, we fall 

 upon what I call the case of inverse orthogonalism. 



This conception will be found to present itself naturally in the 

 course of certain investigations connected with the calculus of 

 sign-progressions suggested by the form of Newton's rule ; and 

 that calculus in its turn leads to a theory of tessellation highly 

 curious in itself, and fruitful of consequences to the calculus of 

 operations and the theory of numbers, furnishing interesting 

 food for thought, or a substitute for the want of it, alike to the 

 analyst at his desk and the fine lady in her boudoir. 



2. In a strictly orthogonal matrix the ?z 2 — l equations resulting 

 from the equal ratios above referred to, on account of the implica- 



ft ^ _L yi 



tions existing between them, really amount to no more than — - — 



n* — n . 



independent conditions, leaving — ^ — of the n 2 terms arbitrary. 



This law, which it would perhaps not be easy to obtain from a 

 direct inspection of the equations, is an instantaneous consequence 

 of the fact that a sum of the squares of n variables may be trans- 

 formed into a sum of squares of n linear functions of the same 

 by means of an orthogonal substitution, — and that, vice versa, 

 such faculty of transformation is sufficient to establish the cha- 



* Communicated by the Author. 



f For a matrix of the order 2 the ratio of each element to its reciprocal 

 in an orthogonal matrix is necessarily +1. This is a case of exception, and 

 may be disregarded. In all other cases the ratio can be varied ad libitum. 



