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larly ascending pyramid, of which the two diagonal planes are 

 termed the planes of separation and symmetry respectively. 

 The former divides the pyramid into two halves, such that 

 no element on the one side of it is the same as that of any 

 block in the other. The plane of symmetry, as the name 

 denotes, divides the pyramid into two exactly similar parts ; 

 it being a rule, that all elements lying in any given line of 

 a square {plafond) parallel to the platie of separation are 

 identical; moreover, the sum of the characteristics is the 

 same, for all elements lying any where in a plane parallel 

 to that of separation. 



All the terms in the final derivee are made up by multi- 

 plying n elements of the pile together, under the sole restric- 

 tion, that no two or more terms of the said product shall 

 lie in any one plane out of the two sets of planes perpendi- 

 cular to the sides of the squares. The sig?i of any such pro- 

 duct is determined by the places of either set of planes 

 parallel to a side of the squares and to one another, in 

 which the elements composing it may be conceived to lie. 



The Author then enters into a disquisition relating to 

 the number of terms which will appear in the final derivee, 

 and concludes this first part with the statement of two 

 general canons, each of which aflfords as many tests for de- 

 termining whether a prepared combination of coefficients 

 can enter into the final derivee of any number of equations 

 as there are units in that number, but so connected as 

 together only to afford double that number, less one of in- 

 dependent conditions. 



The first of these canons refers simply to the number of 

 letters drawn out of each of the given equations, (supposed 

 homogeneous) ; the second to what he proposes to call the 

 weight of every term in the derivee in respect to each of the 

 variables which are to be eliminated. 



The Author subjoins, for the purpose of conveying a more 



