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row one of IL down the right edge of a strip of paper using the same spacing as for the observations. Now place this movable strip alongside the observation vector so that the top element on the paper strip is opposite the top element of the observation vector. Multiply adjacent elements and write the sum of these products at the top of a new column. Now slide the paper strip down t, spaces. Form the indicated inner product as before and write the result in the new column below the previous entry. Continue in this manner until all the observations have been used. Now write row two of MD on a strip of paper and proceed as before. If we continue this process with all the rows of Mn we will get a new vector zn whose elements are linear transformations of the observation vector y, The dimension of z,, is the same as that of y. Similarly form znBl from Zn and &-I . Continuing this process we finally obtain z1 = z which is the desired interaction vector. In all the foregoing we used the normalized contrast matrices; thus the sums of squares are the squares of the elements of z. For hand computation, one might prefer using the unnormalized contrast matrices, since their elements are integers. But then we need a vector of divisors; it is obtained by performing the same operations on a column of ones as on y, except that we use the squares of the elements of the contrast matrices. Then the ith sum of squares equals 2; 2 divided by the corresponding divisor. This method might be called a “paper strip method” for analysis of variance and is similar to paper strip methods used for operations with polynomials. For examples of this, see Lanczos [3] and Pragcr [4]. We require 21J2 l . . t,, locations for storing y and z plus sup(ll , t ‘2, “‘, 1,‘) locations for storilq a row of iV, . The number of multiplicntions rcquircxl is (nli) (r1i + 1). AC~;NO\~LI:DGMI:NTS: The author wishes to thank Dr. A. E. Brxndt for initirlting h i s intcrcst in programming analysis of vwiancc. I It! wishes LO t,h:~& Dr. W. I l . Carter , J r . , and the rcf(lrcbc, for 1~4 pf ul (*oinm(fil ts. Ii 1~:~1~:1~I~:Nc!l:s : 1. GOOI), 1. J. ‘I’hc intcr:~.ction ulgorithm und practical FouricI mdysis. J. lb?/. S,c(utl.s/.. Sot. 113) 20, 2 (1958), 361-372. 2. Gow, I. ,J. The intcractio11 :\.lgori thm :md prnctictll Fouricl mdysis: AI) :~ddcnd~~rn. ,I. Il’o~~. Stat is t . Sot. [I31 $2, 3 (l!KO). 372-375. 3. LANCZOS, CA /1 pphd 4A nah~sis. l’rcM,icc-I1:111, Englcwood Cliffs, N.J., 195G. 4 . ~RAGIX, w’. hlroduction t o Ihsic I~ortrw~ l’royrunttt~i~rg tmd Xu?~rcricul Melds. Bl:lisdcll, M’al tllam, l'hss., 19G5. 5 . YATIS, F. The des ign and an:tlysis of fttctoritll csperimcnts. Imperial Bureau of Soil Science, II:~rper&n, England, 1937.

Algorithm 360: shortest-path forest with topological ordering [H]