Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/135

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CHAP. VIII.]
OF REDUCTION.
119

and in these expressions replace, for simplicity, by , by , &c., we shall have from the three last equations, (1);(2);(3)[errata 1]; and from this system we must eliminate .

Multiplying the second of the above equations by , and the third by , and adding the results to the first, we have [errata 2]. When is made equal to , and therefore to , the first member of the above equation becomes . And when in the same member is made and , it becomes . Hence the result of the elimination of may be expressed in the form (4); and from this equation is to be determined.

Were we now to proceed as in former instances, we should multiply together the factors in the first member of the above equation; but it may be well to show that such a course is not at all necessary. Let us develop the first member of (4) with reference to , the symbol whose expression is sought, we find ; or,; whence we find, ; and developing the second member with respect to and ,


  1. Correction: should be amended to : detail
  2. Correction: should be amended to : detail