Microsoft and Google clearly see each other as mortal enemies. Google dominates in search and online ads; Microsoft is the biggest player in business software. Yet, in consolidating their domains, they seem headed in opposite directions.

Microsoft has made a $44.6 billion bid for Yahoo, a move seen as its effort to challenge the growing power of Google over the Internet. Google, on the other hand, keeps pumping out products that take on Microsoft. The latest is Google Sites, a publishing and collaboration product clearly aimed at Microsoft’s SharePoint.

Are they headed in the right direction? If you believe in classic business strategy, neither is focusing on improving what they do best. Microsoft, a powerhouse in the workplace, seems determined to acquire Internet hipness by purchasing a company that has its own definition of troubles. Google, which makes almost all its money serving up ads in response to Internet searches, has slyly edged its way into the applications market that is the core of Microsoft’s business.

A better move for Microsoft, Randall Stross argued last Sunday in The New York Times, would be to acquire SAP, the German giant of enterprise software. He quoted long-time industry observer Michael Cusumano calling Microsoft’s bid for Yahoo a grab for “an old-style Internet asset, in decline, and at a premium.” Actually, Microsoft and SAP have talked in the past, but they never came to an agreement. Mr. Stross contrasted Microsoft’s latest effort with those of Larry Ellison at Oracle, who has relentlessly acquired companies that add to his company’s database and enterprise strengths.

Google has not followed a coherent pattern either; the company has moved from search to maps to applications. Where it differs from Microsoft is its trust in “the cloud,” using the Internet as a distribution platform for applications, something difficult for Microsoft, which is so heavily invested in traditional software distribution. Even Google Sites, offered as an on-demand application, contrasts sharply with Microsoft’s SharePoint product, which requires software, servers and internal management by the customer. And today, the company introduced Google Health, providing information on health issues.

Following the logic of Mr. Cusumano’s argument, Google may not be playing to its strengths either. The on-demand applications Google has created are adequate, but hardly challengers to Microsoft’s core Office suite. And there is growing concern about the privacy of documents created and stored on Google’s servers. Can companies ever be comfortable with the idea that they don’t have the locks to their most valuable possessions or that they are being scanned for key words so Google can provide relevant ads? Maybe Google should focus on solving the existing challenges in efficient search of video, audio, photos, and other rich media that threaten to overwhelm us.

My biggest worry is that Microsoft is trying to buy relevance. Time-Warner did that in 2000, when executives, worried that they were falling out of fashion, bought AOL, a disastrous acquisition from which the media giant has never recovered. Let’s hope that Microsoft’s push for Yahoo is not a repeat of that desperate -- and often fatal -- need to be loved by the young.